You might think that a field called “asset pricing” would explain the prices of assets. You would be wrong. Instead, it is mostly about asset returns and expected returns.

They’re not that different because a return is simply one price divided by previous one. To actually get to prices, you need to estimate cash flows (or earnings or dividends), and academics, with some exceptions, hate doing that. So we’re left with these weird ratios of prices known as “returns”.

In my post on Lars Peter Hansen, I wrote down an asset pricing model based on a representative consumer utility maximization problem. Using slightly different assumptions, we can justify the so-called Capital Asset Pricing Model or CAPM, which is an older model that holds that returns can be described by \begin{equation} \label{capm} E_t[r_{i,t+1} - r_{f,t+1}] = \beta_i \lambda, \end{equation} where \( r_{f,t+1} \) is the risk-free rate and \( \lambda \) is a number known as the risk premium and \( \beta_i \) is the regression coefficient from a regression of asset returns \( r_{i, t+1} \) on the market return, \( r_{m, t+1} \). (That’s assuming there is a risk free asset. If not, the result changes slightly.) This implies that assets that are more correlated with the market have a higher expected (excess) return, while assets that are uncorrelated or even negatively correlated with the market have a lower expected return, because they provide more insurance against fluctuations in the market.

These days almost every stock picking site lists “beta”, usually the slope of a regression of returns on the S&P 500 index or perhaps a broader market index. An implication of CAPM in the form of equation \( \eqref{capm} \) is that the intercept of that regression, known as Jensen’s alpha or just alpha or abnormal return, is zero.

Ever since this stuff was first proposed, it has been well known that alpha is not zero when you actually run those regressions. (Of course there are ton of econometric disputes in this area. Nobody does empirical research because it is easy and fun.) It’s not zero for individual stocks, but individual stocks are weird and maybe that’s because of noise or other shenanigans.

What’s was more worrying is that the return on somewhat mechanical trading strategies did not have zero alpha: A portfolio of stocks with a high ratio of book value to market value (“value stocks”) has a higher alpha than one with a low such ratio (“growth stocks”). A portfolio of stocks of small companies has a higher alpha than one a portfolio of large company stocks. There are other examples like this, and they suggest that CAPM does not provide a good explanation of the cross section of expected stock returns, or why different stocks have different expected returns.

That’s worrying for the efficient markets hypothesis if CAPM is The True Model. The results of Fama and French (1992) and (1993) suggest that it may not be. Based on the empirical evidence, they propose that expected stock returns are related not just to the stock’s exposure to market risk, but also to two additional factors: The return on a portfolio that is long value stocks and short growth stocks (“HML” or high minus low), and the return on a portfolio that is long small stocks and short big stocks (“SMB” or small minus big). If you do a multivariate regression \begin{equation} r_{i,t+1} = \alpha + \beta_m r_{m,t+1} + \beta_{\mathit{HML}} r_{\mathit{HML},t+1} + \beta_{\mathit{SMB}} r_{\mathit{SMB}, t+1} + \varepsilon_{t+1}, \end{equation} you have an alpha against what is now called the Fama–French 3-factor model. When you let \( r_{i,t+1} \) be returns on portfolios of stocks sorted by either value, size, or both, the resulting 3-factor alphas are a lot closer to zero. Here are the t-statistics:

It’s not perfect, but it’s better than people were able to do before. If Fama–French is the correct model, EMH is in slightly better shape.

Since then a ton of researchers have tried to add factors to the model to better explain the cross section of expected returns, the most widely used being the Carhart momentum factor, to form a 4-factor model. The 3-factor and 4-factor models are the most widely used models in finance for almost any setting where expected and abnormal returns are studied.

There have been many attempts to explain why the value and size factors exist and what explains the risk premia associated with them, i.e. the size of the premium those stocks command. Most of them revolve around hypotheses that the market index does not fully capture the systematic, undiversifiable risk that investors are exposed to. For example, one explanation is that both value and size factors are related to distress risk, the risk of being exposed extra costs associated with financial distress that are not fully captured in the market return measure.

Most papers that propose new factors—someone once claimed that there are 50 factors in the literature explaining returns, but I find that figure quite low—propose some kind of explanation. Some of the arguments build on consumption based asset pricing. One example is a factor relating to takeover risk: the hypothesis is that for companies that are likely to be bought, much of the expected return comes from a potential takeover premium. But takeovers are cyclical and come in waves in a way that you can’t diversify away, so investors have to be rewarded for that risk in addition to market risk (and whatever value and small stock premia represent).

* The photos on that piece are now lost to posterity. The first one is reproduced above, courtesy of mahalanobis. The last one was French with some blonde models. Also, I think an MBA from Rochester costs a lot more than $19.95.

When I first accessed the post, the java math rendering script failed, littering the post with [Math Processing Error] wherever an equation should be. I was wondering if the author was being delightfully sarcastic about how economics really works, but upon reload, I see that was not the case. It would have been wonderfully meta.

I checked out some videos linked from Dimensional's "Fama-French Forum", where Fama discusses these extra "risk factors". The one about small vs large companies makes sense, a small company can go under more easily. But I would have naively expected "value" stocks to be less risky than "growth" stocks. Fama explains that the characteristics of companies given low values in the stock market tend to involve some kind of distress or restructuring, but then it should be the correlated characteristics you focus on. What made it more plausible was when he presented a graph of average return vs standard deviation of returns for different classes of stocks (ranked by book/market value). The class with the highest book/market values also had the highest returns, but the standard deviation within that class was much, much higher than the stocks with a lower book/market ratio. So there could be more uncertainty of returns for individual stocks.